I recently read a charming little paper: Quasi-elliptic integrals and periodic continued fractions, by van der Poorten and Tran. Most of us who have taken a number theory course of some kind learned how to solve Pell’s equation: where
is a nonsquare positive integer. The usual method is to compute the continued fraction
.
One then defines the convergents of by
etcetera.
Then tends to be very small and, if you compute long enough, for some
you will have
.
What van der Poorten and Tran do is to ask what happens if is not an integer, but a polynomial
. Before I get into details, I want to tell you about something gorgeous that I won’t explain at all. Using the methods in their paper, van der Poorten and Trap can discover identities like
Isn’t that pretty?
It turns out that the continued fraction algorithm for is actually much prettier than for integers. Everything should be understood in terms of the curve
cut out by
. This is a curve of genus
, with two points at infinity. (One of these points is the limit of
and the other is the limit of
.) I’ll call these two points
and
. The theory is controlled by the line bundles
. In particular, there are nontrivial solutions to
if and only if the continued fraction is periodic, if and only if
for some
.
Below the fold, I’ll explain what is meant by the continued fraction algorithm for an algebraic function, and tell you some of the other nice results from the paper.
Given any power series in
, we define the continued fraction of
.
Define . Set
and define
by
. Then set
and
. Continuing in this way, we get a sequence
of polynomials, a sequence
of power series, and a continued fraction
.
We can also define the convergents as before; they do converge to
in the sense that each ratio
agrees with
to a higher order than the ratio does.
In particular, suppose that is a polynomial of the form
.
Then is a power series in
:
So we can define the continued fraction of .
We keep the notations ,
,
and
from above.
I’ll explain just one key idea from the paper. Let’s think about the zeroes and poles of . Since
is a polynomial in
, its only poles are at
, and it has a pole of the same order at both
‘s. So, other than
, the function
has the same poles as
. Then
has zeroes at the poles of
.
That’s what happens away from . Suppose that
has a pole of order
at
, and a zero of order
at
. Then
has a pole of order
at both
‘s. The difference
has a zero of order
at
and a pole of order
at
. So
has a pole of order
at
and a zero of order
at
.
Summing up the last two paragraphs, let the poles of be
and let the zeroes be
. Then the poles of
are
, for some
and some
and the zeroes are
. (Here
,
and
are supported away from
.) In other words, there is a sequence of positive integers
and a sequence of divisors
such that the poles of
are
while the zeroes are
.
Note that is the degree of
. Note also that
in the Picard group, so
.
It’s not too hard to work out what happens if the coefficients of are chosen generically. The first
has degree
and all the other
are
. A bit of effort checks that
has degree
(exercise!), so all of
have degree
and, in fact,
in the Picard group. You may remember that a generic divisor of degree
has a unique effective representative in PIcard;
is that unique representative. So, we have just found an explicit way to write down an arithmetic progression in
, where
is a hyperelliptic curve.
Of course, the fun comes in the nongeneric case. In that case, the can skip around. It’s really fun when
is torsion in the Picard group or, in other words, when there is a unit
in the coordinate ring of
. Then, eventually, the sequence in Picard will repeat. It turns out, when this happens, the corresponding approximation
gives your unit!
There are plenty of other ideas in the paper. What is the analogue of the result that the are the best approximations to
? The
are all of the form
: how do we relate the polynomials
and
to the divisors
? And how did I come up with that integral above? All this and more, so read the paper!
I don’t understand what is meant by
[Y(t)] :=\sum_{i=0}^k Y_k t_k
Are some of those k’s perhaps supposed to be i’s or something?
I assume he means that you take all the terms from the power series where t has a non-negative exponent.
I’m pretty sure Kenny’s right, so I corrected the post moving the subscript to a superscript. David if I did this wrong please change it back.
Thanks for the fix, but you missed the other typo: the summation variable is
, not
. Fixed now.
Kenny’s English summary is completely correct.